We note here that Einstein gravity is a parity invariant theory. In order to get a theory with a non-zero \(c_L\) , one needs to add a gravitational Chern-Simons (GCS) term to the Einstein-Hilbert action. The resulting theory, called Topologically Massive Gravity [1]}, thus has an action
\( S_{\text{TMG}} &=& S_{\text{EH}} + S_{\text{GCS}} + S_{\text{bdy}} \\ &=& \int d^3x \sqrt{-g} \left\lbrace \mathcal {R} + \frac{1}{2\mu } \varepsilon ^{\lambda \mu \nu }\Gamma ^\rho _{\, \lambda \sigma } \left( \partial _\mu \Gamma ^\sigma _{\, \rho \nu } + \frac{2}{3} \Gamma ^\sigma _{\, \mu } \Gamma ^_{\, \nu \rho }\right) \right\rbrace + S_{\text{bdy}} \nonumber \) 
